Finite Group Databasebeta

← #126993

Subgroups — #126993

Order 512.

880 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1465C212yesmaximal
2#13785C211yesmaximal, characteristic
3#7189C214yesmaximal, complemented
4C22×C4×D4C2211yescharacteristic
5C2×#407—22no—
6C2×#390C412yes—
7C2×C4×C4⋊C4C411yescharacteristic
8C2×#376C411yescharacteristic
9#1465—24no—
10#1465C418yescomplemented
11C2×#324C2214yes—
12C2×#323—22no—
13#1437—22no—
14C4⋊#95C2212yes—
15C22⋊#95—22no—
16C23×D4C2×C411yescharacteristic
17C24×C4D412yes—
18C22×M16—42no—
19C2×C4×D4—42no—
20C2×C4×D4—42no—
21C2×C4×D4C2×C414yes—
22C2×C4×D4D412yes—
23C2×C4×D4Q812yes—
24C22×C4⋊C4C2×C412yes—
25C22×C4⋊C4C2311yescharacteristic
26C22×C22⋊C4—24no—
27C22×C22⋊C4D412yes—
28C22×C42—21no—
29C22×C42C2×C411yescharacteristic
30C22×C42C2×C412yes—
31#407—28no—
32#407—44no—
33C2×C22⋊C8—24no—
34C2×C22⋊C8—42no—
35#390—24no—
36#390—24no—
37C4×C4⋊C4—42no—
38#376—42no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—24no—
43#324—24no—
44#324C2×C418yes—
45#323—28no—
46C25C22⋊C412yes—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—41no—
50C22×D4—41no—
51C22×D4—42no—
52C22×D4C4212yes—
53C22×D4C4⋊C412yes—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—24no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—84no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C414yes—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C4⋊C414yes—
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D412yes—
66C2×M16—48no—
67C2×M16—88no—
68C22×C8—44no—
69C4×D4—44no—
70C4×D4—44no—
71C4×D4—44no—
72C4×D4—44no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4M1614yescomplemented
79C2×C4⋊C4C22×C412yes—
80C2×C4⋊C4C2×D411yescharacteristic
81C2×C4⋊C4C2×Q811yescharacteristic
82C2×C22⋊C4—216no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—84no—
86C2×C22⋊C4—84no—
87C2×C22⋊C4C22⋊C414yes—
88C2×C22⋊C4C4⋊C414yes—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—42no—
93C2×C42—42no—
94C2×C42—42no—
95C2×C42—82no—
96C2×C42C4212yes—
97C2×C42C4⋊C412yes—
98C2×C42M1614yescomplemented
99C22⋊C8—28no—
100C22⋊C8—44no—
101C22⋊C8—48no—
102#95—42no—
103#95—42no—
104#95—42no—
105#95—84no—
106C24—24no—
107C24—22no—
108C24—22no—
109C24—24no—
110C24—42no—
111C24—44no—
112C24—42no—
113C24#9514yes—
114C24C2×C22⋊C411yescharacteristic
115C2×D4—24no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—44no—
123C2×D4—44no—
124C2×D4—44no—
125C2×D4—44no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—42no—
140C22×C4—48no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—84no—
155C22×C4—84no—
156C22×C4—84no—
157C22×C4—88no—
158C22×C4#9512yes—
159C22×C4#9512yes—
160C22×C4#9912yes—
161C22×C4#9912yes—
162C22×C4C2×C4211yescharacteristic
163C22×C4C2×C22⋊C412yes—
164C22×C4C2×C4⋊C411yescharacteristic
165C22×C4C2×C4⋊C412yes—
166C22×C4C2×M1612yes—
167C22×C4C2×M1612yes—
168M16—816no—
169C2×C8—48no—
170C2×C8—88no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C22⋊C4—44no—
177C22⋊C4—44no—
178C22⋊C4—816no—
179C42—24no—
180C42—24no—
181C42—44no—
182C42—44no—
183C42—84no—
184C23—22no—
185C23—21no—
186C23—21no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—44no—
191C23—44no—
192C23—48no—
193C23—48no—
194C23—44no—
195C23—42no—
196C23—42no—
197C23—44no—
198C23—44no—
199C23—82no—
200C23—82no—
201C23—84no—
202C23—84no—
203C23C2×#9511yescharacteristic
204C23#40712yes—
205C23C2×#9911yescharacteristic
206C23C2×#9911yescharacteristic
207C23#41312yes—
208D4—48no—
209D4—44no—
210D4—44no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—48no—
226C2×C4—48no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—48no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—84no—
240C2×C4—84no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—84no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4#34214yes—
250C2×C4C4×M1612yes—
251C2×C4#42311yescharacteristic
252C2×C4#42311yescharacteristic
253C8—88no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—48no—
263C22—44no—
264C22—44no—
265C22—44no—
266C22—84no—
267C22—88no—
268C22—84no—
269C22—84no—
270C22#146512yes—
271C22#146512yes—
272C22#172811yescharacteristic
273C22C2×#34211yescharacteristic
274C22#174211yescharacteristic
275C22#212712yes—
276C22#213411yescharacteristic
277C22#213511yescharacteristic
278C4—24no—
279C4—22no—
280C4—21no—
281C4—21no—
282C4—42no—
283C4—42no—
284C4—42no—
285C4—84no—
286C4—84no—
287C2—22no—
288C2—21no—
289C2—21no—
290C2—84no—
291C2#791111yescharacteristic
292C2#1153312yes—
293C2#1153512yes—
294C2#1258211yescharacteristic
295C2#1258711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #7189C2semidirect
10. #1465C4semidirect
78. C2×C4⋊C4M16semidirect
98. C2×C42M16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#14652
#137852
#71892